
Our baseline data set \(\mathcal {D}\)  includes the Planck 2018 [1]}, TT,TE, and EE data for low-\(\ell \)  (`lowl_TT', `lowl_EE') and high-\(\ell \)  (`highl_TTTEEE') with the full set of nuisance parameters.
It also includes the late-universe constraints: the BAO-only likelihood (`bao_boss_dr12') from BOSS DR12 (\(z = 0.38, 0.51, 0.61\) )[2]} and the small-z BAO likelihood (`bao_smallz_2014') including data from the 6dF (\(z = 0.106\) )[3]} and MGS (\(z = 0.15\) ) [4]} catalogs, as well as the PANTHEON [5]} supernova likelihood (`Pantheon').

The data set \(\mathcal {H}\)  is chosen to test the Hubble tension, it consists of the latest measurement of the intrinsic magnitude of supernovae \(M_b=-19.253\pm 0.027\)  by the SHOES collaboration [6]} , which we implement as a Gaussian likelihood for this parameter.

The data set \(\mathcal {S}\)  is chosen to test the \(S_8\)  tension. It includes the \(3\times 2 pt\)  weak lensing and galaxy clustering analyses by KiDS-1000x{2dFLenS+BOSS} [7]} and DES-Y3 [8]} which obtain \(S_8=0.766^{+0.020}_{-0.014}\)  and \(S_8=0.775^{+0.026}_{-0.024}\) , which we implement as simple asymmetric Gaussian likelihoods for \(S_8\) . For quantifying the “Gaussian Tension” we also combine the two \(S_8\)  measurements and their positive \(1\sigma \)  ranges to \(S_8^{\rm direct} = 0.769 \pm 0.016\) . In addition, the data set \(S\)  includes the Planck lensing [1]} likelihood (`Planck_lensing').

